Mappings that preserve Sidon sets in R Colin C. Graham

0. Introduction and statement of results. A closed subset E of R is a Helson set if for every f~Co(E) there exists a Fourier transform gEA(R) such that g[e=f. A Sidon set is a countable Helson set. Every function h: R ~ R that induces (by composition) an automorphism of A(R) clearly maps Helson sets to Helson sets; such h are exactly the affine functions [BH]. Since the union of two Helson sets is again a Helson set (that is due to Drury and Varopoulos; see [GM, chapter 2] for a proof), every continuous function f : R-~R whose graph consists of a finite number of straight line segments (possibly of infinite length) maps Helson sets to Helson sets.

It is therefore reasonable to ask whether functions f that map Helson sets to Helson sets have graphs consiting of a finite number of straight line segments. Theo- rem 1 shows that if the homeomorphism f : R-~R maps Sidon sets to Sidon sets, then the graph o f f consists of a countable number of straight line segments having a finite number of distinct slopes. Theorem 1 and its proof appear in Section 1. In Section 2 we give an example that shows that the condition of Theorem 1 can- not be improved globally. Whether Sidon set-preserving homeomorphisms are locally piecewise a/fine is unknown. Section 3 contains our concluding remarks,

1. Proof of the main result, We will use the following definitions. A continuous function f : R ~ R is c.p.a. (countabIy pieeewise affine) if the set of x such t b a t f is affine in a neighborhood of x is dense in R. The c.p.a, function f has afinite num- ber of slopes if the slopes of the segments of the grar, h o f f belong to a finite set.

Theorem 1. Let f: R ~ R be a homeomorphism such that f (E) is a Sidon set whenever E is a Sidon set. Then f is e.p.a, with a fni te number of slopes.

We will prove Theorem 1 by proving a slightly stronger result (Theorem 2). For that we need the following.

It is well-known that a Sidon or Helson set cannot contain a sequence of subsets of the form A,+B, where Card A , = C a r d B,=n for n=>l. A set E is dissociate

218 Colin C. Graham

if for all n _ -> 1, distinct elements Xl . . . . , x,,EE and choices e~, ..., e~6 {0, _+ 1, _2}, ejxj = 0 only if all the zi are zero. A countable closed dissociate set is necessarily

Sidon. I f E~, Ez, ... is a sequence of disjoint subsets of R, we say that w Ej is an independent union if Gp EjwGPUkejEk={O}. If a countable union E of dis- sociate sets is an independent union and has at most one cluster point, then E is a Sidon set. For proofs see [LR] or [GM].

Theorem 2. (i) Let g: R-~R be a continuous function that is not c.p.a. Then

there exist sequences {An}, {B,} of subsets of R such that for all n, Card A, = Card B, =n and g ( A , + B , ) is dissociate and such that ~ g ( A , + B , ) is an independent union with at most one cluster point.

(ii) Let g: R + R be c.p.a, with an infinite number of distinct slopes. Then there exist sequences {A,} and {B,} with the same properties of (i).

How Theorem 2 implies Theorem 1. In both cases (i) and (ii) of Theorem 2 we set g = f - L Let E = g ( A , + B , ) - , where the A, and B, are given by Theorem 2 (i) or (ii), assuming that f is not c.p.a, with a finite number of slopes. Then E is a Sidon set. But f ( E ) = U ~ ( A , + B , ) - contains arbitrarily large squares, so f can- not preserve Sidon sets. Theorem 1 thus follows from Theorem 2.

We shall need the following Lemma for the proof of Theorem 2 (i).

Lemma 3. Let g: R ~ R be a continuous. Let m>=l. Suppose that there exist integers {ni, j}i'n,j=a such that the function h(ul, vl, ..., u m, v,,)=Z~,j=lni, jg(ui+v j) from R 2m to R is constant on an open set. Then either all the ni, j are zero, or g agrees

with an affine function on an open set.

Proof. Let e > 0 and I~={(xl . . . . , x2m)CR~m: tXjl<e for all j}. Suppose that i p r p h is constant on (ut, vl, ..., Urn, Vm)+I,, and that not all n~,j=O. Let {fk} be any

bounded approximate identity for L~(R) such that each fk is twice continuously differentiable and has support in 1~/2. Then the function defined by

hk(U 1 . . . . . Vm) = Z } , j F l l , j ( f k * g ) ( U l - - ~ - I ) j )

I I is constant in X=(u~ . . . . . v,,)+I~/~ and is twice continuously differentiable. �9 0 2 hk C~

Suppose that ni, i r Then b v ~ = V in X. But

02 hk ~,j * It U OViOU~- .~n~.~(A*g) ' (u~+v3 = n~i(A g) (~+vj) .

Therefore fk * g is affine in the interval I = (u; + v~.-- e/2, u; + vj. + e/2). Since fk * g ~ g uniformly in 1~/2, g is affine in I~/~. The Lemma follows.

Mappings that preserve Sidon sets in R 219

Proof of Theorem 2 (i). Suppose that g is not c.p.a. Then there exists an open interval (a, b ) r such that g is not affine on any open non-empty subset of (a, b). Without loss of generality, we may assume that a < 0 < b (translate) and that b > 1 (change scale). Let A1 = B I = {1}.

Suppose that n => 1 and that sets A1, B~ . . . . , An, B n have been found such that, for all l~_m<=n,

(1) (2) (3)

Let For

AmWBmC=[O, 2-m], Card Am=Card B,,=m; g(Am+Bm) is dissociate; and

U~'=I g(Ak+Bk) is an independent union.

a . A B H = U L 1 G P ( T U ~ = , g ( k+ k)). n + l all sets q = {n~j}~,j=, of integers, let

Z(q) = {(u 1, vl, . . . , un+l, v,+a)E[0,2-n-l]~'+=: Znl jg(ui+vl) -H}.

Since H is countable, Lemma 3 implies that X(t/) is a union of closed sets, each having no interior. Let X be the union of the X(q) as r/ ranges over all subsets {hi. ~: i, j = 1 . . . . , n + 1 } of (n + 1) 2 integers. Then X is also of first (Baire) category, so there exists (u~, v1 . . . . , u ,+l , vn+0E[0, 2-"-1]~"+~\X. Then, for A,+I = {ul, ..., u,+l} and B ,+ l = {v l . . . . , Vn+l }, we see that (1)--(3) hold for m = n + l . Now Theorem 2 (i) follows.

Proof of Theorem 2 (ii). Suppose that g is c.p.a, having the form g(x)=akX+bk in the interval [l k, rk], k = 1, 2 . . . . . Suppose also that {ak} contains an infinite set of distinct numbers.

Let {ak(j)}7=l be such that the ak(j) are distinct. Without loss of generality we may assume that lk(j+l)<lk(j) for j=>l (we may need to replace g(x) by g(-x)) . I f lim Ik~j)=l exists, we may assume that I=0 . By passing to a subsequence {ak(j) } we may assume that for all choices m=>l and n j={0 , _+1, __+2} for l<=j<=m.

;.n (4) ~1 njak(J) : 0 only if nl . . . . . ti m = 0.

(That is merely the routine matter of extracting a dissociate subsequence from

{"~cJ3.) We now consider sums of the form S=Z:j=lni , j(ak.)(ui+vy ) +bi), where the

1 n~,j are chosen f rom {0, +1 , +_2}, the u, f rom [lk(o,-~(lk(i)+rkti))] , and the vj f rom [0, 1 ~(rk(i)--lk(o) ]. The set X of (ul, Vl . . . . , urn, Vm) such that S = 0 has no interior, for if X had interior then on varying the u~ and vj, we would conclude

that Y~i nt, j ak (i)=0 for each j , thus contradicting (4). The argument now proceedes exactly as in the proof of Part (i). The remaining details are left to the reader.

220 Colin C. Graham

2. An example. The following example shows that we cannot conclude that a mapping preserving Helson sets is affine on neighborhoods of +oo and - ~ .

We define f : R--,-R as follows.

I -~x+n 2n~=x<=2n+l

(5) f(x)----- _~ _ n _ l 2 n + l - < _ x N 2 n + 2 for n = l , 2 . . . .

otherwise.

We claim thatfpreserves Helson sets in R. It will suffice to show that for each >0 there is a constant C such that the Helson constant o f f (E) is at most C if the

Helson constant of E is at most a; here E ranges over compact Helson sets. Any compact Helson set has the form E=U~=I U~=IE,,] where E 0 , j = ( - ~ , 2 ] c ~ E and E,,~En[2n+(j-1)/8, 2n+j/8] for 1~=j~16 and n = l , 2 , . . . . Applica- tion of the Saucer principle [GM, 11.4] shows that, for all j ' s and all m ~ 1, the

A "* algebras (U,=l [2n+( j -1 ) ]8 , 2n+j/8]) and A([ ( j -1) /8 , j /8] • {2 . . . . ,2m}) are is omorphic. For 1 ~j_-< 8, fmaps [ ( j - 1) 8, j/8] + 2n linearly to [ ( j - 1 )/16, j/16] + 2n; for 8< j~16 , f maps [(j-1)/8,j/8]+2n linearly to [3(j-1)/16,3j/16]+2n-1. The Saucer Principle again applies: A(U'~[a(j-1)/16, aj/16]+2n-b) and A([a(j-1)/61, aj/16]• n--l , m, m}) are isomorphic, where a = l , b = 0 for l < j ~ 8 and a=3, b---I for 8<j~16 . Therefore for all l~-j_-<16, and m_->l, a ( f ( U ~ E,,j))~=Cla (U~ E,,~). Bythe union theorem for Helson sets, a(f(E))~-C.

3. Remarks. (i) This paper was stimulated by a question asked by R. S. Pierce at a seminar in Honolulu.

(ii) If t h e f o f Theorem 1 is only assumed to preserve compact Sidon sets, t h e n f is c.p.a, and the restrictions o f f to each compact interval have but a finite number of slopes (the number may increase with the size of the interval).

(iii) A "compact" version of the example of Section 2 could be given if we could answer the following affirmatively.

Do there exist sequences x,>=O, O~=a,~=b, with lim b ,= l imx~=0 such that for every Sidon set E~= 1 (Ec~ [a,, b,])+x~ is a Sidon set?

(iv) Our Theorem 1, combined with the Lemma of Beufling and Helson [BH, P. 121] yields immediately their Corollary: if f: R-~ R preserves Fourier transforms, then f is affine.

(v) Our definition of "c.p.a." is a weak one. One might replace it with the fol- lowing: f i s c.p.a, if for all L>0 , the sum of the lengths of the maximal intervals (a, b) in [ - L , L] such that f is affine in (a, b) equals 2L. We do not know if Helson set preserving maps have that property.

Mappings that preserve Sidon sets in R 221

References

BH BEURLING, A. and HELSON, H., Fourier--Stieltjes transforms with bounded powers. Math. Scand. 1 (1958), 120--126.

GM GRAHAM, C. C. and McGErIE~, O. C., Essays in Commutative Harmonic Analysis. New York: Springer-Verlag, 1979.

LR L6PEZ, J. and Ross, K. A., Sidon Sets. New York: Marcel Dekker, 1975.

Received January 16, 1980 Department of Mathematics Northwestern University EVANSTON, IL 60201 USA